1 00:00:00,100 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:21,122 --> 00:00:22,080 PROFESSOR: Let's begin. 9 00:00:26,540 --> 00:00:31,190 Today we're going to continue the discussion on Ito calculus. 10 00:00:31,190 --> 00:00:33,840 I briefly introduced you to Ito's lemma last time, 11 00:00:33,840 --> 00:00:38,510 but let's begin by reviewing it and stating it 12 00:00:38,510 --> 00:00:39,910 in a slightly more general form. 13 00:00:42,570 --> 00:00:45,790 Last time what we did was we did the quadratic variation 14 00:00:45,790 --> 00:00:51,830 of Brownian motion, Brownian process. 15 00:00:57,140 --> 00:01:01,160 We defined the Brownian process, Brownian motion, 16 00:01:01,160 --> 00:01:11,700 and then showed that it has quadratic variation, which 17 00:01:11,700 --> 00:01:22,380 can be written in this form-- dB square is equal to dt. 18 00:01:22,380 --> 00:01:33,350 And then we used that to show the simple form of Ito's lemma, 19 00:01:33,350 --> 00:01:38,080 which says that if f is a function on the Brownian 20 00:01:38,080 --> 00:01:48,820 motion, then d of f is equal to f prime of dB_t 21 00:01:48,820 --> 00:01:56,990 plus f double prime of dt. 22 00:01:56,990 --> 00:02:03,730 This additional term was a characteristic of Ito calculus. 23 00:02:03,730 --> 00:02:06,405 In classical calculus we only have this term, 24 00:02:06,405 --> 00:02:08,509 but we have this additional term. 25 00:02:08,509 --> 00:02:10,389 And if you remember, this happened exactly 26 00:02:10,389 --> 00:02:13,690 because of this quadratic variation. 27 00:02:13,690 --> 00:02:17,100 Let's review it, and let's do it in a slightly more general 28 00:02:17,100 --> 00:02:18,530 form. 29 00:02:18,530 --> 00:02:20,630 As you know, we have a function f 30 00:02:20,630 --> 00:02:24,690 depending on two variables, t and x. 31 00:02:24,690 --> 00:02:30,830 Now we're interested in-- we want to evaluate 32 00:02:30,830 --> 00:02:35,956 our information on the function f(t, B_t). 33 00:02:38,890 --> 00:02:41,120 The second coordinate, we're planning 34 00:02:41,120 --> 00:02:43,966 to put in the Brownian motion there. 35 00:02:43,966 --> 00:02:45,590 Then again, let's do the same analysis. 36 00:02:45,590 --> 00:02:52,530 Can we describe d of f in terms of these differentiations? 37 00:02:52,530 --> 00:02:58,610 To do that, deflect this, let me start from Taylor expansion. 38 00:03:04,460 --> 00:03:16,080 f at a point t plus delta t, x plus delta x by Taylor 39 00:03:16,080 --> 00:03:31,400 expansion for two variables is f of t of x plus partial of f 40 00:03:31,400 --> 00:03:39,068 over partial of t at t comma x of delta t plus... 41 00:03:42,980 --> 00:03:44,510 x. 42 00:03:44,510 --> 00:03:47,651 That's the first-order terms. 43 00:03:47,651 --> 00:03:49,150 Then we have the second-order terms. 44 00:04:27,560 --> 00:04:31,440 Then the third-order terms, and so on. 45 00:04:31,440 --> 00:04:34,700 That's just Taylor expansion. 46 00:04:34,700 --> 00:04:37,030 If you look at it, we have a function f. 47 00:04:37,030 --> 00:04:39,370 We want to look at the difference of f when we change 48 00:04:39,370 --> 00:04:41,661 the first variable a little bit and the second variable 49 00:04:41,661 --> 00:04:42,830 a little bit. 50 00:04:42,830 --> 00:04:44,984 We start from f of t of x. 51 00:04:44,984 --> 00:04:47,400 In the first-order terms, you take the partial derivative, 52 00:04:47,400 --> 00:04:50,350 so take del f over del t, and then multiply 53 00:04:50,350 --> 00:04:52,200 by the t difference. 54 00:04:52,200 --> 00:04:54,350 Second term, you take the partial derivative 55 00:04:54,350 --> 00:04:57,420 with respect to the second variable-- partial f 56 00:04:57,420 --> 00:05:02,470 over partial x-- and then multiply by del x. 57 00:05:02,470 --> 00:05:05,560 That much is enough for classical calculus. 58 00:05:05,560 --> 00:05:08,000 But then, as we have seen before, 59 00:05:08,000 --> 00:05:09,750 we ought to look at the second-order term. 60 00:05:09,750 --> 00:05:14,030 So let's first write down what it is. 61 00:05:14,030 --> 00:05:16,370 That's exactly what happened in Taylor expansion, 62 00:05:16,370 --> 00:05:17,140 if you remember. 63 00:05:17,140 --> 00:05:20,130 If you don't remember, just believe me. 64 00:05:20,130 --> 00:05:23,856 This 1 over 2 times, take the second derivatives, partial. 65 00:05:26,400 --> 00:05:29,100 Let's write it in terms of-- yes? 66 00:05:29,100 --> 00:05:31,806 AUDIENCE: [INAUDIBLE] 67 00:05:34,464 --> 00:05:36,978 PROFESSOR: Oh, yeah, you're right. 68 00:05:36,978 --> 00:05:37,946 Thank you. 69 00:05:45,700 --> 00:05:46,350 Is it good now? 70 00:05:50,060 --> 00:05:52,150 Let's write it as dt, all these deltas. 71 00:05:56,592 --> 00:05:57,830 I'll just write like that. 72 00:05:57,830 --> 00:06:00,270 I'll just not write down t and x. 73 00:06:00,270 --> 00:06:05,980 And what we have is f plus del f over del t dt plus del 74 00:06:05,980 --> 00:06:10,810 f over del x dx plus the second-order terms. 75 00:06:37,290 --> 00:06:39,990 The only important terms-- first of all, these terms 76 00:06:39,990 --> 00:06:42,620 are important. 77 00:06:42,620 --> 00:06:44,660 But then, if you want to use x equals B of 78 00:06:44,660 --> 00:06:49,780 t-- so if you're now interested in f t comma B of t. 79 00:06:49,780 --> 00:06:55,970 Or more generally, if you're interested in f t plus dt, 80 00:06:55,970 --> 00:07:03,285 f B_t plus d of B_t, then these terms are important. 81 00:07:03,285 --> 00:07:07,720 If you subtract f of t of B_t, what you get 82 00:07:07,720 --> 00:07:11,460 is these two terms. 83 00:07:11,460 --> 00:07:16,916 Del f over del t dt plus del f over del 84 00:07:16,916 --> 00:07:19,970 x-- I'm just writing this as a second variable 85 00:07:19,970 --> 00:07:22,035 differentiation-- at dB_t. 86 00:07:25,880 --> 00:07:28,550 And then the second-order terms. 87 00:07:28,550 --> 00:07:32,605 Instead of writing it all down, dt square is insignificant, 88 00:07:32,605 --> 00:07:37,580 and dt comma-- dt times dB_t also is insignificant. 89 00:07:37,580 --> 00:07:39,910 But the only thing that matters will be this one. 90 00:07:39,910 --> 00:07:45,000 This is dB_t square, which you saw is equal to dt. 91 00:07:48,920 --> 00:07:52,990 From the second-order term, we'll have this term surviving. 92 00:07:52,990 --> 00:08:01,160 1 over 2 partial f over partial x second derivative, of dt. 93 00:08:01,160 --> 00:08:04,010 That's it. 94 00:08:04,010 --> 00:08:05,860 If you rearrange it, what we get is 95 00:08:05,860 --> 00:08:18,259 partial f over partial t plus 1/2 this plus-- 96 00:08:18,259 --> 00:08:19,550 and that's the additional term. 97 00:08:25,150 --> 00:08:28,620 If you ask me why these terms are not important 98 00:08:28,620 --> 00:08:33,150 and this term is important, I can't really say it rigorously. 99 00:08:33,150 --> 00:08:36,929 But if you think about dB_t square equals dt, then d times 100 00:08:36,929 --> 00:08:39,210 B_t is kind of like square root of dt. 101 00:08:39,210 --> 00:08:40,860 It's not a good notation, but if you 102 00:08:40,860 --> 00:08:45,870 do that-- these two terms are significantly smaller than dt 103 00:08:45,870 --> 00:08:48,030 because you're taking a power of it. 104 00:08:48,030 --> 00:08:51,905 dt square becomes a lot smaller than dt, dt to the 3/2 105 00:08:51,905 --> 00:08:54,630 is a lot smaller than dt. 106 00:08:54,630 --> 00:08:59,601 But this one survives because it's equal to dt here. 107 00:08:59,601 --> 00:09:01,225 That's just the high-level description. 108 00:09:05,530 --> 00:09:08,320 That's a slightly more sophisticated form 109 00:09:08,320 --> 00:09:09,470 of Ito's lemma. 110 00:09:09,470 --> 00:09:12,370 Let me write it down here. 111 00:09:12,370 --> 00:09:14,513 And let's just fix it now. 112 00:09:18,441 --> 00:09:48,880 If f of t of B_t-- that's d of f is equal to-- Any questions? 113 00:09:58,610 --> 00:10:01,360 Just remember, from the classical calculus term, 114 00:10:01,360 --> 00:10:05,385 we're only adding this one term there. 115 00:10:05,385 --> 00:10:05,884 Yes? 116 00:10:05,884 --> 00:10:09,580 AUDIENCE: Why do we have x there? 117 00:10:09,580 --> 00:10:15,400 PROFESSOR: Because the second variable is supposed to be x. 118 00:10:15,400 --> 00:10:18,316 I don't want to write down partial derivative with respect 119 00:10:18,316 --> 00:10:21,240 to a Brownian motion here because it doesn't look good. 120 00:10:24,190 --> 00:10:26,490 It just means, take the partial derivative with respect 121 00:10:26,490 --> 00:10:28,910 to the second term. 122 00:10:28,910 --> 00:10:33,340 So just view this as a function f of t of x, 123 00:10:33,340 --> 00:10:42,080 evaluate it, and then plug in x equal B_t in the end, 124 00:10:42,080 --> 00:10:44,250 because I don't want to write down partial B_t here. 125 00:10:51,234 --> 00:10:51,900 Other questions? 126 00:11:11,810 --> 00:11:27,436 Consider a stochastic process X of t such that d of X 127 00:11:27,436 --> 00:11:32,030 is equal to mu times d of t plus sigma times d of B_t. 128 00:11:35,360 --> 00:11:38,160 This is almost like a Brownian motion, 129 00:11:38,160 --> 00:11:39,720 but you have this additional term. 130 00:11:39,720 --> 00:11:41,221 This is called a drift term. 131 00:11:46,130 --> 00:11:53,245 Basically, this happens if X_t is equal to mu*t plus sigma 132 00:11:53,245 --> 00:11:55,710 of B_t. 133 00:11:55,710 --> 00:11:57,714 Mu and sigma are constants. 134 00:12:01,387 --> 00:12:02,970 From now on, what we're going to study 135 00:12:02,970 --> 00:12:08,390 is stochastic process of this type, whose difference 136 00:12:08,390 --> 00:12:12,608 can be written in terms of drift term and the Brownian motion 137 00:12:12,608 --> 00:12:13,107 term. 138 00:12:16,100 --> 00:12:18,165 We want to do a slightly more general form 139 00:12:18,165 --> 00:12:21,780 of Ito's lemma, where we want f of t of X_t here. 140 00:12:25,502 --> 00:12:27,085 That will be the main object of study. 141 00:12:31,593 --> 00:12:34,105 I'll finally state the strongest Ito's lemma 142 00:12:34,105 --> 00:12:35,244 that we're going to use. 143 00:12:44,924 --> 00:12:54,370 f is some smooth function and X_t is a stochastic process 144 00:12:54,370 --> 00:12:56,151 like that. 145 00:12:56,151 --> 00:12:57,095 X_t satisfies... 146 00:13:06,230 --> 00:13:08,770 where B_t is a Brownian motion. 147 00:13:08,770 --> 00:13:17,210 Then df of t, X_t can be expressed 148 00:13:17,210 --> 00:13:38,510 as-- it's just getting more and more complicated. 149 00:13:38,510 --> 00:13:41,580 But it's based on this one simple principle, really. 150 00:13:41,580 --> 00:13:45,110 It all happened because of quadratic variation. 151 00:13:45,110 --> 00:13:49,800 Now I'll show you why this form deviates from this form when 152 00:13:49,800 --> 00:13:58,320 we replace B to an X. 153 00:13:58,320 --> 00:14:03,400 Remember here all other terms didn't matter, 154 00:14:03,400 --> 00:14:08,595 that the only term that mattered was partial square of f... 155 00:14:08,595 --> 00:14:11,888 of dx square. 156 00:14:17,490 --> 00:14:30,990 To prove this, note that df is partial f over partial t 157 00:14:30,990 --> 00:14:37,066 dt plus partial f over partial x d of X_t 158 00:14:37,066 --> 00:14:42,775 plus 1/2 of d of x squared. 159 00:14:45,490 --> 00:14:48,970 Just exactly the same, but I've replaced the dB-- previously, 160 00:14:48,970 --> 00:14:52,885 what we had dB, I'm replacing to dX. 161 00:14:52,885 --> 00:14:58,280 Now what changes is dX_t can be written like that. 162 00:14:58,280 --> 00:15:03,580 If you just plug it in, what you get here 163 00:15:03,580 --> 00:15:14,010 is partial f over partial x mu dt plus sigma of dB_t. 164 00:15:14,010 --> 00:15:19,680 Then what you get here is 1/2 of partials 165 00:15:19,680 --> 00:15:23,630 and then mu plus sigma dB_t square. 166 00:15:26,620 --> 00:15:31,250 Out of those three terms here we get mu square dt square 167 00:15:31,250 --> 00:15:37,590 plus 2 times mu sigma d mu dB plus sigma square dB square. 168 00:15:37,590 --> 00:15:40,370 Only this was survives, just as before. 169 00:15:40,370 --> 00:15:42,970 These ones disappear. 170 00:15:42,970 --> 00:15:45,180 And then you just collect the terms. 171 00:15:45,180 --> 00:15:48,690 So dt-- there's one dt here. 172 00:15:48,690 --> 00:15:55,673 There's mu times that here, and that one will become a dt. 173 00:15:55,673 --> 00:16:00,308 It's 1/2 of sigma square partial square... 174 00:16:00,308 --> 00:16:01,300 of dt. 175 00:16:01,300 --> 00:16:04,770 And there's only one dB_t term here. 176 00:16:04,770 --> 00:16:14,312 Sigma-- I made a mistake, sigma. 177 00:16:25,080 --> 00:16:27,480 This will be a form that you'll use the most, 178 00:16:27,480 --> 00:16:32,710 because you want to evaluate some stochastic process-- 179 00:16:32,710 --> 00:16:36,150 some function that depends on time 180 00:16:36,150 --> 00:16:37,431 and that stochastic process. 181 00:16:37,431 --> 00:16:39,180 You want to understand the difference, df. 182 00:16:42,930 --> 00:16:44,660 The X would have been written in terms 183 00:16:44,660 --> 00:16:47,000 of a Brownian motion and a drift term, 184 00:16:47,000 --> 00:16:50,090 and then that's the Ito lemma for you. 185 00:16:50,090 --> 00:16:51,680 But if you want to just-- if you just 186 00:16:51,680 --> 00:16:56,420 see this for the first time, it just looks too complicated. 187 00:16:56,420 --> 00:16:59,460 You don't understand where all the terms are coming from. 188 00:16:59,460 --> 00:17:01,150 But in reality, what it's really doing 189 00:17:01,150 --> 00:17:05,359 is just take this Taylor expansion. 190 00:17:05,359 --> 00:17:08,170 Remember these two classical terms, 191 00:17:08,170 --> 00:17:11,190 and remember that there's one more term here. 192 00:17:11,190 --> 00:17:12,780 You can derive it if you want to. 193 00:17:18,990 --> 00:17:21,030 Really try to know where it all comes from. 194 00:17:21,030 --> 00:17:28,140 It all started from this one fact, quadratic variation, 195 00:17:28,140 --> 00:17:32,940 because that made some of the second derivative survive, 196 00:17:32,940 --> 00:17:34,850 and because of those, you get these kind 197 00:17:34,850 --> 00:17:35,931 of complicated terms. 198 00:17:39,180 --> 00:17:39,680 Questions? 199 00:17:51,165 --> 00:17:53,444 Let's do some examples. 200 00:17:53,444 --> 00:17:54,110 That's too much. 201 00:18:02,390 --> 00:18:05,260 Sorry, I'm going to use it a lot, so let me record it. 202 00:18:49,671 --> 00:18:54,590 Example number one. 203 00:18:54,590 --> 00:19:04,270 Let f of x be equal to x square, and then you 204 00:19:04,270 --> 00:19:07,160 want to compute d of f at B_t. 205 00:19:13,590 --> 00:19:16,726 I'll give you three minutes just to try a practice. 206 00:19:16,726 --> 00:19:18,030 Did you manage to do this? 207 00:19:25,610 --> 00:19:26,875 It's a very simple example. 208 00:19:32,400 --> 00:19:37,740 Assume it's just the function of two variables, 209 00:19:37,740 --> 00:19:40,010 but it doesn't depend on t. 210 00:19:40,010 --> 00:19:44,030 You don't have to do that, but let me just do that. 211 00:19:44,030 --> 00:19:45,755 Partial f over partial t is 0. 212 00:19:49,040 --> 00:19:52,970 Partial f over partial x is equal to 2x, 213 00:19:52,970 --> 00:20:01,580 and the second derivative equal to 2 at t, x. 214 00:20:01,580 --> 00:20:08,010 Now we just plug in t comma B_t, and what 215 00:20:08,010 --> 00:20:11,345 you have is mu equals 0, sigma equals 1, 216 00:20:11,345 --> 00:20:13,011 if you want to write it in this formula. 217 00:20:19,460 --> 00:20:25,940 What you're going to have is 2 times B_t of dB_t 218 00:20:25,940 --> 00:20:27,490 plus 1 over 2 times 2dt. 219 00:20:30,322 --> 00:20:31,266 If you write it down. 220 00:20:34,570 --> 00:20:36,815 You can either use these parameters 221 00:20:36,815 --> 00:20:41,200 and just plug in each of them to figure it out. 222 00:20:41,200 --> 00:20:43,130 Or a different way to do it is really 223 00:20:43,130 --> 00:20:45,340 write down, remember the proof. 224 00:20:45,340 --> 00:20:48,490 This is partial f over partial t dt 225 00:20:48,490 --> 00:20:58,350 plus partial f over partial x dx plus 1/2-- remember this one. 226 00:20:58,350 --> 00:21:00,045 And x is dB_t here. 227 00:21:04,190 --> 00:21:09,100 That one is 0, that one was 2x, so 2B_t dB_t. 228 00:21:09,100 --> 00:21:11,872 Use it one more time, so you get dt. 229 00:21:20,600 --> 00:21:21,280 Make sense? 230 00:21:24,160 --> 00:21:26,855 Let's do a few more examples. 231 00:22:03,150 --> 00:22:06,876 And you want to compute d of f at t comma B of t. 232 00:22:11,280 --> 00:22:13,810 Let's do it this time. 233 00:22:13,810 --> 00:22:19,440 Again, partial f over partial t dt plus partial f 234 00:22:19,440 --> 00:22:23,500 over partial x dB_t. 235 00:22:23,500 --> 00:22:24,820 That's the first-order terms. 236 00:22:24,820 --> 00:22:28,740 The second-order term is 1/2 partial square f 237 00:22:28,740 --> 00:22:35,728 over partial x square of dB_t square, which is equal to dt. 238 00:22:43,140 --> 00:22:43,640 Let's do it. 239 00:22:43,640 --> 00:22:48,720 Partial f over partial t, you get mu times f. 240 00:22:48,720 --> 00:22:51,016 This one is just equal to mu times f. 241 00:22:53,720 --> 00:22:55,220 Maybe I'm going too quick. 242 00:22:55,220 --> 00:23:02,510 Mu times e to the mu t plus dx, dt. 243 00:23:02,510 --> 00:23:05,593 Partial f over partial x is sigma times e 244 00:23:05,593 --> 00:23:12,120 to the mu t plus dx, and then dB_t 245 00:23:12,120 --> 00:23:15,190 plus-- if you take the second derivative, 246 00:23:15,190 --> 00:23:17,545 you do that again, what you get is 247 00:23:17,545 --> 00:23:25,872 1/2, and then sigma square times e to the mu t plus dx, dt. 248 00:23:25,872 --> 00:23:26,372 Yes? 249 00:23:26,372 --> 00:23:28,012 AUDIENCE: In the original equation that you just wrote, 250 00:23:28,012 --> 00:23:29,816 isn't it 1/2 times sigma squared, 251 00:23:29,816 --> 00:23:31,784 and then the second derivative? 252 00:23:31,784 --> 00:23:33,854 Up there. 253 00:23:33,854 --> 00:23:34,520 PROFESSOR: Here? 254 00:23:34,520 --> 00:23:35,470 AUDIENCE: Yes. 255 00:23:35,470 --> 00:23:36,094 PROFESSOR: 1/2? 256 00:23:36,094 --> 00:23:37,940 AUDIENCE: Times sigma squared. 257 00:23:37,940 --> 00:23:40,510 PROFESSOR: Oh, sigma-- OK, that's a good question. 258 00:23:40,510 --> 00:23:44,070 But that sigma is different. 259 00:23:44,070 --> 00:23:46,250 That's if you plug in X_t here. 260 00:23:46,250 --> 00:23:50,790 If you plug in X_t where X_t is equal to mu 261 00:23:50,790 --> 00:23:57,170 prime dt plus sigma prime d of B_t, 262 00:23:57,170 --> 00:23:59,500 then that sigma prime will become a sigma prime square 263 00:23:59,500 --> 00:24:02,020 here. 264 00:24:02,020 --> 00:24:04,124 But here the function is mu and sigma, 265 00:24:04,124 --> 00:24:05,540 so maybe it's not a good notation. 266 00:24:05,540 --> 00:24:07,383 Let me use a and b here instead. 267 00:24:11,170 --> 00:24:13,890 The sigma here is different from here. 268 00:24:13,890 --> 00:24:15,950 AUDIENCE: Yeah, that makes a lot more sense. 269 00:24:15,950 --> 00:24:18,826 PROFESSOR: If you replace a and b, 270 00:24:18,826 --> 00:24:23,310 but I already wrote down all mu's and sigma's. 271 00:24:23,310 --> 00:24:25,495 That's a good point, actually. 272 00:24:25,495 --> 00:24:26,995 But that's when you want to consider 273 00:24:26,995 --> 00:24:29,250 a general stochastic process here 274 00:24:29,250 --> 00:24:31,276 other than Brownian motion. 275 00:24:31,276 --> 00:24:33,030 But here it's just a Brownian motion, 276 00:24:33,030 --> 00:24:35,376 so it's the most simple form. 277 00:24:35,376 --> 00:24:36,375 And that's what you get. 278 00:24:39,300 --> 00:24:45,940 Mu plus 1/2 sigma square-- and these are just all f itself. 279 00:24:45,940 --> 00:24:48,130 That's the good thing about exponential. 280 00:24:48,130 --> 00:24:51,939 f times dt plus sigma times d of B_t. 281 00:25:03,930 --> 00:25:04,570 Make sense? 282 00:25:14,180 --> 00:25:17,896 And there's a reason I was covering this example. 283 00:25:17,896 --> 00:25:22,140 It's because-- let's come back to this question. 284 00:25:22,140 --> 00:25:35,110 You want to model stock price using Brownian motion, Brownian 285 00:25:35,110 --> 00:25:40,760 process, S of t. 286 00:25:40,760 --> 00:25:43,040 But you don't want S_t to be a Brownian motion. 287 00:25:43,040 --> 00:25:46,760 What you want is a percentile difference 288 00:25:46,760 --> 00:25:51,400 to be a Brownian motion, so you want this percentile difference 289 00:25:51,400 --> 00:25:58,505 to behave like a Brownian motion with some variance. 290 00:26:03,350 --> 00:26:11,500 The question was, is S_t equal to e to the sigma times B of t 291 00:26:11,500 --> 00:26:12,745 in this case? 292 00:26:12,745 --> 00:26:16,260 And I already told you last time that no, it's not true. 293 00:26:16,260 --> 00:26:18,360 We can now see why it's not true. 294 00:26:18,360 --> 00:26:20,880 Take this function, S_t equals e to the sigma 295 00:26:20,880 --> 00:26:24,990 B_t, that's exactly where mu is equal to 0 here. 296 00:26:24,990 --> 00:26:30,590 What we got here was d of S_t, in this case, is equal to-- mu 297 00:26:30,590 --> 00:26:36,430 is 0, so we get 1/2 of sigma square times dt plus sigma 298 00:26:36,430 --> 00:26:37,274 times d of B_t. 299 00:26:40,180 --> 00:26:44,670 We originally were targeting sigma times dB_t, 300 00:26:44,670 --> 00:26:47,890 but we got this additional term which we 301 00:26:47,890 --> 00:26:51,112 didn't want in the first hand. 302 00:26:51,112 --> 00:26:52,570 In other words, we have this drift. 303 00:27:01,455 --> 00:27:03,080 I wasn't really clear in the beginning, 304 00:27:03,080 --> 00:27:05,850 but our goal was to model stock price 305 00:27:05,850 --> 00:27:10,920 where the expected value is 0 at all times. 306 00:27:10,920 --> 00:27:12,970 Our guess was to take e to the sigma B_t, 307 00:27:12,970 --> 00:27:14,890 but it turns out that in this case 308 00:27:14,890 --> 00:27:16,870 we have a drift, if you just take 309 00:27:16,870 --> 00:27:19,570 natural e to the sigma of B_t. 310 00:27:19,570 --> 00:27:21,230 To remove that drift, what you can do 311 00:27:21,230 --> 00:27:23,350 is subtract that term somehow. 312 00:27:23,350 --> 00:27:26,570 If you can get rid of that term then you can see, 313 00:27:26,570 --> 00:27:30,120 if you add this mu to be minus 1 over 2 sigma square, 314 00:27:30,120 --> 00:27:31,340 you can remove that term. 315 00:27:34,460 --> 00:27:35,650 That's why it doesn't work. 316 00:27:35,650 --> 00:27:47,570 So instead use S of t equals e to the minus 1 over 2 sigma 317 00:27:47,570 --> 00:27:52,532 square t plus sigma of B_t. 318 00:27:58,330 --> 00:28:02,850 That's the geometric Brownian motion without drift. 319 00:28:02,850 --> 00:28:05,820 And the reason it has no drift is because of that. 320 00:28:05,820 --> 00:28:07,300 If you actually do the computation, 321 00:28:07,300 --> 00:28:08,390 the dt term disappears. 322 00:28:28,911 --> 00:28:29,410 Question? 323 00:28:35,580 --> 00:28:39,000 So far we have been discussing differentiation. 324 00:28:39,000 --> 00:28:40,706 Now let's talk about integration. 325 00:28:40,706 --> 00:28:41,206 Yes? 326 00:28:41,206 --> 00:28:47,122 AUDIENCE: Could you we do get this solution as [INAUDIBLE]. 327 00:28:47,122 --> 00:28:50,573 Could you also describe what it means? 328 00:28:50,573 --> 00:28:55,996 What does it mean, this solution of B_t? 329 00:28:55,996 --> 00:28:58,461 Does that mean if we have a sample path B_t, 330 00:28:58,461 --> 00:29:01,440 then we could get a sample path for S_t? 331 00:29:01,440 --> 00:29:03,670 PROFESSOR: Oh, what this means, yes. 332 00:29:03,670 --> 00:29:07,360 Whenever you have the B_t value, just at each time 333 00:29:07,360 --> 00:29:10,050 take the exponential value. 334 00:29:10,050 --> 00:29:13,460 Because-- why we want to express this in terms of a Brownian 335 00:29:13,460 --> 00:29:14,940 motion is, for Brownian motion we 336 00:29:14,940 --> 00:29:17,030 have a pretty good understanding. 337 00:29:17,030 --> 00:29:21,280 It's a really good process you understand fairly well, 338 00:29:21,280 --> 00:29:25,200 and you have good control on it. 339 00:29:25,200 --> 00:29:28,840 But the problem is you want to have a process whose percentile 340 00:29:28,840 --> 00:29:31,990 difference behaves like a Brownian motion. 341 00:29:31,990 --> 00:29:34,340 And this gives you a way of describing it 342 00:29:34,340 --> 00:29:37,290 in terms of Brownian motion, as an exponential function of it. 343 00:29:43,390 --> 00:29:46,555 Does that answer your question? 344 00:29:46,555 --> 00:29:48,330 AUDIENCE: Right, distribution means 345 00:29:48,330 --> 00:29:50,720 that if we have a sample path B_t, 346 00:29:50,720 --> 00:29:54,544 that would be the corresponding sample path for S of t? 347 00:29:54,544 --> 00:29:58,260 Is it a pointwise evaluation? 348 00:29:58,260 --> 00:30:00,270 PROFESSOR: That's a good question, actually. 349 00:30:00,270 --> 00:30:02,950 Think of it as a pointwise evaluation. 350 00:30:02,950 --> 00:30:07,400 That is not always correct, but for most 351 00:30:07,400 --> 00:30:09,150 of the things that we will cover, 352 00:30:09,150 --> 00:30:13,120 it's safe to think about it that way. 353 00:30:13,120 --> 00:30:16,690 But if you think about it path-wise all the time, 354 00:30:16,690 --> 00:30:18,680 eventually it fails. 355 00:30:18,680 --> 00:30:20,187 But that's a very advanced topic. 356 00:30:32,130 --> 00:30:33,740 So what this question is, basically 357 00:30:33,740 --> 00:30:37,350 B_t is a probability space, it's a probability distribution 358 00:30:37,350 --> 00:30:39,390 over paths. 359 00:30:39,390 --> 00:30:43,060 For this equation, if you just look at it, it looks right, 360 00:30:43,060 --> 00:30:44,650 but it doesn't really make sense, 361 00:30:44,650 --> 00:30:46,964 because B_t-- if it's a probability distribution, what 362 00:30:46,964 --> 00:30:47,630 is e to the B_t? 363 00:30:50,450 --> 00:30:52,530 Basically, what it's saying is B_t 364 00:30:52,530 --> 00:30:55,210 is a probability distribution over paths. 365 00:30:55,210 --> 00:30:58,700 If you take omega according to-- a path according 366 00:30:58,700 --> 00:31:02,890 to the Brownian motion sample probability distribution, 367 00:31:02,890 --> 00:31:08,230 and for this path it's well defined, this function. 368 00:31:08,230 --> 00:31:13,700 So the probability density function of this path 369 00:31:13,700 --> 00:31:16,910 is equal to the probability density function of e 370 00:31:16,910 --> 00:31:19,435 to the whatever that is in this distribution. 371 00:31:24,410 --> 00:31:26,300 Maybe it confused you more. 372 00:31:26,300 --> 00:31:30,009 Just consider this as some path, some well-defined function, 373 00:31:30,009 --> 00:31:31,550 and you have a well-defined function. 374 00:31:39,490 --> 00:31:40,853 Integral definition. 375 00:31:44,000 --> 00:31:46,510 I will first give you a very, very stupid definition 376 00:31:46,510 --> 00:31:49,340 of integration. 377 00:31:49,340 --> 00:32:00,293 We say that we define F as the integration... 378 00:32:12,270 --> 00:32:25,574 if d of F is equal to f dB_t plus-- we 379 00:32:25,574 --> 00:32:27,365 define it as an inverse of differentiation. 380 00:32:30,200 --> 00:32:34,860 Because differentiation is now well-defined-- 381 00:32:34,860 --> 00:32:39,690 we just defined integration as the inverse of it, 382 00:32:39,690 --> 00:32:42,170 just as in classical calculus. 383 00:32:46,030 --> 00:32:47,810 So far, it doesn't have that good meaning, 384 00:32:47,810 --> 00:32:51,160 other than being an inverse of it, 385 00:32:51,160 --> 00:32:52,710 but at least it's well-defined. 386 00:32:52,710 --> 00:32:54,780 The question is, does it exist? 387 00:32:54,780 --> 00:32:57,445 Given f and g, does it exist, does integration always exist, 388 00:32:57,445 --> 00:32:58,060 and so on. 389 00:32:58,060 --> 00:32:59,690 There's lots of questions to ask, 390 00:32:59,690 --> 00:33:02,570 but at least this is some definition. 391 00:33:02,570 --> 00:33:11,760 And the natural question is, does there exist a Riemannian 392 00:33:11,760 --> 00:33:12,740 sum type description? 393 00:33:25,430 --> 00:33:28,580 That means-- if you remember how we defined integral 394 00:33:28,580 --> 00:33:47,740 in calculus, you have a function f, integration 395 00:33:47,740 --> 00:33:58,570 of f from a to b according to the Riemannian sum description 396 00:33:58,570 --> 00:34:04,660 was, you just chop the interval into very fine pieces, 397 00:34:04,660 --> 00:34:07,730 a_0, a_1, a_2, a_3, dot, dot, dot-- 398 00:34:07,730 --> 00:34:16,750 and then sum the area of these boxes, and take the limit. 399 00:34:16,750 --> 00:34:20,570 And this is the limit of Riemannian sums. 400 00:34:26,420 --> 00:34:33,920 Slightly more, if you want, it's the limit as n goes to infinity 401 00:34:33,920 --> 00:34:40,790 of the function 1 over n times the sum of i equal zero to t-- 402 00:34:40,790 --> 00:34:48,587 I'll just call it 0 to b-- f of t*b over n minus f of t minus 1 403 00:34:48,587 --> 00:34:51,461 over n. 404 00:34:51,461 --> 00:34:52,734 Does this ring a bell? 405 00:35:03,787 --> 00:35:04,287 Question? 406 00:35:04,287 --> 00:35:05,162 AUDIENCE: [INAUDIBLE] 407 00:35:10,780 --> 00:35:13,832 PROFESSOR: No, you're right. 408 00:35:13,832 --> 00:35:17,115 Good point, no we don't. 409 00:35:17,115 --> 00:35:17,615 Thanks. 410 00:35:22,570 --> 00:35:26,010 Does integral defined in this way 411 00:35:26,010 --> 00:35:31,390 have this Riemannian sum type description, is the question. 412 00:35:31,390 --> 00:35:33,110 So keep that in mind. 413 00:35:33,110 --> 00:35:36,660 I will come back to this point later. 414 00:35:36,660 --> 00:35:41,839 In fact, it turns out to be a very deep question 415 00:35:41,839 --> 00:35:43,630 and very important question, this question, 416 00:35:43,630 --> 00:35:48,850 because if you remember like I hope you remember, 417 00:35:48,850 --> 00:35:51,860 in the Riemannian sum, it didn't matter which point you 418 00:35:51,860 --> 00:35:54,610 took in this interval. 419 00:35:54,610 --> 00:35:56,910 That was the whole point. 420 00:35:56,910 --> 00:35:58,780 You have the function. 421 00:35:58,780 --> 00:36:02,430 In the interval a_i to a_(i+1), you take any point 422 00:36:02,430 --> 00:36:07,610 in the middle and make a rectangle according to that 423 00:36:07,610 --> 00:36:08,642 point. 424 00:36:08,642 --> 00:36:10,350 And then, no matter which point you take, 425 00:36:10,350 --> 00:36:12,740 when you go to the limit, you had exactly the same sum 426 00:36:12,740 --> 00:36:14,190 all the time. 427 00:36:14,190 --> 00:36:16,560 That's how you define the limit. 428 00:36:16,560 --> 00:36:20,960 But what's really interesting here 429 00:36:20,960 --> 00:36:24,090 is that it's no longer true. 430 00:36:24,090 --> 00:36:26,000 If you take the left point all the time, 431 00:36:26,000 --> 00:36:28,110 and you take the right point all the time, 432 00:36:28,110 --> 00:36:30,570 the two limits are different. 433 00:36:30,570 --> 00:36:33,020 And again, that's due to the quadratic variation, 434 00:36:33,020 --> 00:36:38,980 because that much of variance can accumulate over time. 435 00:36:38,980 --> 00:36:42,560 That's the reason we didn't start with Riemannian sum type 436 00:36:42,560 --> 00:36:44,450 definition of integral. 437 00:36:44,450 --> 00:36:47,490 But I'll just make one remark. 438 00:36:47,490 --> 00:37:01,220 Ito integral is the limit of Riemannian sums 439 00:37:01,220 --> 00:37:08,240 when always take the leftmost point of each interval. 440 00:37:16,700 --> 00:37:20,670 So you chop down this curve at-- the time interval 441 00:37:20,670 --> 00:37:23,106 into pieces, and for each rectangle, 442 00:37:23,106 --> 00:37:25,230 pick the leftmost point, and use it as a rectangle. 443 00:37:30,722 --> 00:37:31,680 And you take the limit. 444 00:37:31,680 --> 00:37:33,570 That will be your Ito integral defined. 445 00:37:33,570 --> 00:37:37,070 It will be exactly equal to this thing, the inverse of our Ito 446 00:37:37,070 --> 00:37:38,754 differentiation. 447 00:37:38,754 --> 00:37:40,170 I won't be able to go into detail. 448 00:37:40,170 --> 00:37:44,080 What's more interesting is instead, 449 00:37:44,080 --> 00:37:47,170 what happens if you take the rightmost point all the time, 450 00:37:47,170 --> 00:37:51,680 you get an equivalent theory of calculus. 451 00:37:51,680 --> 00:37:53,380 It's just like Ito's calculus. 452 00:37:53,380 --> 00:37:57,160 It looks really, really similar and it's coherent itself, 453 00:37:57,160 --> 00:37:59,170 so there is no logical flaw in it. 454 00:37:59,170 --> 00:38:01,260 It all makes sense, but the only difference 455 00:38:01,260 --> 00:38:03,984 is instead of a plus in the second-order term, 456 00:38:03,984 --> 00:38:04,650 you get minuses. 457 00:38:07,500 --> 00:38:09,940 Let me just make this remark, because it's just 458 00:38:09,940 --> 00:38:15,350 a theoretical part, this thing, but I think it's really cool. 459 00:38:15,350 --> 00:38:22,150 Remark-- there's this and equivalent version. 460 00:38:22,150 --> 00:38:24,430 Maybe equivalent is not the right word, 461 00:38:24,430 --> 00:38:26,820 but a very similar version of Ito 462 00:38:26,820 --> 00:38:33,000 calculus such that basically, what 463 00:38:33,000 --> 00:38:38,510 it says is d B_t square is equal to minus dt. 464 00:38:38,510 --> 00:38:40,320 Then that changed a lot of things. 465 00:38:40,320 --> 00:38:44,510 But this part, it's not that important. 466 00:38:44,510 --> 00:38:48,740 Just cool stuff. 467 00:38:48,740 --> 00:38:53,410 Let's think about this a little bit more, this fact. 468 00:38:53,410 --> 00:38:55,970 Taking the leftmost point all the time 469 00:38:55,970 --> 00:38:59,820 means if you want to make a decision for your time 470 00:38:59,820 --> 00:39:05,630 interval-- so at time t of i and time t of i plus 1, 471 00:39:05,630 --> 00:39:08,760 let's say it's the stock price. 472 00:39:08,760 --> 00:39:14,590 You want to say that you had so many stocks in this time 473 00:39:14,590 --> 00:39:16,370 interval. 474 00:39:16,370 --> 00:39:20,400 Let's say you had so many stocks in this time interval 475 00:39:20,400 --> 00:39:23,430 according to the values between this and this. 476 00:39:23,430 --> 00:39:26,660 In real world, your only choice you have is you 477 00:39:26,660 --> 00:39:30,700 have to make the decision at time t of i. 478 00:39:30,700 --> 00:39:33,580 Your choice cannot depend on the future time. 479 00:39:33,580 --> 00:39:36,650 You can't suddenly say, OK, in this interval the stock 480 00:39:36,650 --> 00:39:38,420 price increased a lot, so I'll assume 481 00:39:38,420 --> 00:39:42,930 that I had a lot of stocks in this interval. 482 00:39:42,930 --> 00:39:46,190 In this interval, I knew it was going to drop, 483 00:39:46,190 --> 00:39:50,060 so I'll just take the rightmost interval. 484 00:39:50,060 --> 00:39:52,410 I'll assume that I only had this many stock. 485 00:39:52,410 --> 00:39:53,460 You can't do that. 486 00:39:53,460 --> 00:39:56,850 Your decision has to be based on the leftmost point, 487 00:39:56,850 --> 00:39:58,630 because the time. 488 00:39:58,630 --> 00:40:00,880 You can't see the future. 489 00:40:00,880 --> 00:40:04,690 And the reason Ito's calculus works well in our setting is 490 00:40:04,690 --> 00:40:09,380 because of this fact, because it has inside it the fact that you 491 00:40:09,380 --> 00:40:10,820 cannot see the future. 492 00:40:10,820 --> 00:40:16,310 Every decision is made based on the leftmost time. 493 00:40:16,310 --> 00:40:18,845 If you want to make a decision for your time interval, 494 00:40:18,845 --> 00:40:21,700 you have to do it in the beginning. 495 00:40:21,700 --> 00:40:27,240 That intuition is hidden inside of the theory, 496 00:40:27,240 --> 00:40:29,390 and that's why it works so well. 497 00:40:29,390 --> 00:40:33,450 Let me reiterate this part a little bit more. 498 00:40:33,450 --> 00:40:36,500 It's the definition of these things 499 00:40:36,500 --> 00:40:39,540 where you're only allowed to-- at time t, 500 00:40:39,540 --> 00:40:42,230 you're only allowed to use the information up to time t. 501 00:40:54,390 --> 00:41:26,010 Definition: delta t is an adapted process-- sorry, 502 00:41:26,010 --> 00:41:29,730 adapted to another stochastic process X_t-- 503 00:41:29,730 --> 00:41:38,920 if for all values of time variables 504 00:41:38,920 --> 00:41:48,110 delta t depends only on X_0 up to X_t. 505 00:41:50,930 --> 00:41:53,360 There's a lot of vague statements inside here, 506 00:41:53,360 --> 00:41:55,120 but what I'm trying to say is just 507 00:41:55,120 --> 00:41:59,947 assume X is the Brownian motion underlying stock price. 508 00:41:59,947 --> 00:42:00,905 Your stock is changing. 509 00:42:04,500 --> 00:42:06,305 You want to come up with a strategy, 510 00:42:06,305 --> 00:42:08,270 and you want to say that mathematically 511 00:42:08,270 --> 00:42:11,280 this strategy makes sense. 512 00:42:11,280 --> 00:42:13,220 And what it's saying is if your strategy 513 00:42:13,220 --> 00:42:17,050 makes your decision at time t is only 514 00:42:17,050 --> 00:42:19,840 based on the past values of your stock price, 515 00:42:19,840 --> 00:42:23,750 then that's an adapted process. 516 00:42:23,750 --> 00:42:26,590 This defines the processes that are reasonable, 517 00:42:26,590 --> 00:42:28,540 that cannot see future. 518 00:42:28,540 --> 00:42:31,030 And these are all-- in terms of strategy, 519 00:42:31,030 --> 00:42:34,400 if delta_t is a portfolio strategy, 520 00:42:34,400 --> 00:42:37,262 these are the only meaningful strategies that you can use. 521 00:42:40,240 --> 00:42:42,740 And because of what I said before, because we're always 522 00:42:42,740 --> 00:42:45,580 taking the leftmost point, adaptive 523 00:42:45,580 --> 00:42:50,440 processes just also fit very well with Ito's calculus. 524 00:42:50,440 --> 00:42:53,050 They'll come into play altogether. 525 00:42:55,670 --> 00:42:56,655 Just a few examples. 526 00:43:10,445 --> 00:43:13,740 First, a very stupid example. 527 00:43:13,740 --> 00:43:15,290 X_t is adapted to X_t. 528 00:43:20,170 --> 00:43:23,230 Of course, because at time, X_t really 529 00:43:23,230 --> 00:43:26,710 depends on only X_t, nothing else. 530 00:43:26,710 --> 00:43:36,060 Two, X_(t+1) is not adapted to X_t. 531 00:43:36,060 --> 00:43:37,580 This is maybe a little bit vague, 532 00:43:37,580 --> 00:43:41,412 so we'll call it Y_t equals X_(t+1). 533 00:43:44,090 --> 00:43:49,240 Y_t is the value at t plus 1, and it's not based 534 00:43:49,240 --> 00:43:50,850 on the values up to time t. 535 00:43:50,850 --> 00:43:52,600 Just a very artificial example. 536 00:43:56,405 --> 00:44:03,178 Another example, delta t equals minimum... 537 00:44:06,136 --> 00:44:07,122 is adapted. 538 00:44:21,419 --> 00:44:23,180 And I'll let you think about it. 539 00:44:23,180 --> 00:44:24,850 The fourth is quite interesting. 540 00:44:24,850 --> 00:44:27,540 Suppose T is fixed, some large integer, 541 00:44:27,540 --> 00:44:30,140 or some large real number. 542 00:44:30,140 --> 00:44:44,170 Then you let delta t to be the maximum where X of s, where... 543 00:44:50,600 --> 00:44:51,370 It's not adapted. 544 00:44:58,790 --> 00:44:59,455 What is this? 545 00:44:59,455 --> 00:45:02,020 This means at time T, I'm going to take at it 546 00:45:02,020 --> 00:45:08,850 this value, the maximum of all value 547 00:45:08,850 --> 00:45:11,340 inside this part, the future. 548 00:45:11,340 --> 00:45:13,469 This refers to the future. 549 00:45:13,469 --> 00:45:14,635 It's not an adapted process. 550 00:45:21,637 --> 00:45:22,220 Any questions? 551 00:45:25,290 --> 00:45:28,340 Now we're ready to talk about the properties 552 00:45:28,340 --> 00:45:31,190 of Ito's integral. 553 00:45:31,190 --> 00:45:34,340 Let's quickly review what we have. 554 00:45:34,340 --> 00:45:38,380 First, I defined Ito's lemma-- that means differentiation 555 00:45:38,380 --> 00:45:41,250 in Ito calculus. 556 00:45:41,250 --> 00:45:45,080 Then I defined integration using differentiation-- integration 557 00:45:45,080 --> 00:45:48,020 was an inverse operation of the differentiation. 558 00:45:48,020 --> 00:45:50,500 But this integration also had an alternative description 559 00:45:50,500 --> 00:45:53,260 in terms of Riemannian sums, where 560 00:45:53,260 --> 00:45:58,650 you're taking just the leftmost point as the reference 561 00:45:58,650 --> 00:46:01,700 point for each interval. 562 00:46:01,700 --> 00:46:04,370 And then, as you see, this naturally 563 00:46:04,370 --> 00:46:08,090 had this concept of using the leftmost point. 564 00:46:08,090 --> 00:46:12,180 And to abstract that concept, we've 565 00:46:12,180 --> 00:46:15,660 come up with this adapted process, very natural process, 566 00:46:15,660 --> 00:46:17,710 which is like the real-life procedures, 567 00:46:17,710 --> 00:46:20,900 real-life strategies we can think of. 568 00:46:20,900 --> 00:46:22,700 Now let's see what happens when you 569 00:46:22,700 --> 00:46:25,442 take the integral of adapted processes. 570 00:46:25,442 --> 00:46:27,870 Ito integral has really cool properties. 571 00:46:59,540 --> 00:47:03,000 The first thing is about normal distribution. 572 00:47:03,000 --> 00:47:08,840 B_t has normal distribution of 0 up to t. 573 00:47:08,840 --> 00:47:11,170 So your Brownian motion at time t 574 00:47:11,170 --> 00:47:13,780 has normal distribution with 0, t. 575 00:47:13,780 --> 00:47:17,090 That means if your stochastic process is some constant times 576 00:47:17,090 --> 00:47:23,540 B of t, of course, then you have 0 and c square t. 577 00:47:23,540 --> 00:47:26,780 It's still a normal variable. 578 00:47:26,780 --> 00:47:28,770 That means if you integrate, that's 579 00:47:28,770 --> 00:47:31,130 the integration of some sigma. 580 00:47:39,878 --> 00:47:42,058 That's the integration of sigma of dB_t. 581 00:47:46,680 --> 00:47:49,280 If sigma is a fixed constant, when 582 00:47:49,280 --> 00:47:52,830 you take the Ito integral of sigma times 583 00:47:52,830 --> 00:47:55,200 dB_t, this constant, at each time 584 00:47:55,200 --> 00:47:58,210 you get a normal distribution. 585 00:47:58,210 --> 00:48:00,550 And this is like saying the sum of normal distribution 586 00:48:00,550 --> 00:48:02,330 is also normal distribution. 587 00:48:02,330 --> 00:48:04,090 It has this hidden fact, because integral 588 00:48:04,090 --> 00:48:06,980 is like sum in the limit. 589 00:48:06,980 --> 00:48:10,456 And this can be generalized. 590 00:48:10,456 --> 00:48:18,560 If delta t is a process depending only on the time 591 00:48:18,560 --> 00:48:27,660 variable-- so it does not depend on the Brownian motion-- then 592 00:48:27,660 --> 00:48:35,810 the process X of t equals the integration of delta t dB_t 593 00:48:35,810 --> 00:48:50,420 has normal distribution at all time, just like this. 594 00:48:50,420 --> 00:48:52,580 We don't know the exact variance yet; 595 00:48:52,580 --> 00:48:55,280 the variance will depend on the sigmas. 596 00:48:55,280 --> 00:48:57,334 But still, it's like a sum of normal variables, 597 00:48:57,334 --> 00:48:58,750 so we'll have normal distribution. 598 00:49:03,490 --> 00:49:05,360 In fact, it just gets better and better. 599 00:49:10,140 --> 00:49:14,950 The second fact is called Ito isometry. 600 00:49:14,950 --> 00:49:15,930 That was cool. 601 00:49:15,930 --> 00:49:17,136 Can we compute the variance? 602 00:49:29,611 --> 00:49:30,110 Yes? 603 00:49:30,110 --> 00:49:31,466 AUDIENCE: Can you put that board up? 604 00:49:31,466 --> 00:49:32,132 PROFESSOR: Sure. 605 00:49:34,630 --> 00:49:35,970 AUDIENCE: Does it go up? 606 00:49:35,970 --> 00:49:37,900 PROFESSOR: This one doesn't go up. 607 00:49:37,900 --> 00:49:40,070 That's bad. 608 00:49:40,070 --> 00:49:41,240 I wish it did go up. 609 00:49:49,020 --> 00:49:52,060 This has a name called Ito isometry. 610 00:49:56,740 --> 00:49:58,890 Can be used to compute the variance. 611 00:49:58,890 --> 00:50:01,640 B_t is a Brownian motion, delta t 612 00:50:01,640 --> 00:50:03,480 is adapted to a Brownian motion. 613 00:50:09,050 --> 00:50:17,610 Then the expectation of your Ito integral-- 614 00:50:17,610 --> 00:50:21,600 that's the Ito integral of your adapted process. 615 00:50:21,600 --> 00:50:25,440 That's the variance-- we take the square of it-- 616 00:50:25,440 --> 00:50:29,847 is equal to something cool. 617 00:50:36,180 --> 00:50:38,456 The square just comes in. 618 00:50:38,456 --> 00:50:39,500 Quite nice, isn't it? 619 00:50:44,520 --> 00:50:48,400 I won't prove it, but let me tell you why. 620 00:50:48,400 --> 00:50:50,300 We already saw this phenomenon before. 621 00:50:50,300 --> 00:50:51,960 This is basically quadratic variation. 622 00:50:58,000 --> 00:50:59,560 And the proof also uses it. 623 00:50:59,560 --> 00:51:03,160 If you take delta s equals to 1-- sorry, 624 00:51:03,160 --> 00:51:09,680 I was using Korean-- 1 at all time, then what we have is 625 00:51:09,680 --> 00:51:13,490 here you get a Brownian motion, B_t. 626 00:51:13,490 --> 00:51:19,530 So on the left you get like expectation of B_t square, 627 00:51:19,530 --> 00:51:21,525 and on the right, what you get is t. 628 00:51:24,440 --> 00:51:27,445 Because when delta s is equal to 1 629 00:51:27,445 --> 00:51:30,260 at all time, when you have to get from 0 to t you get t, 630 00:51:30,260 --> 00:51:32,730 and you have t on the right hand side. 631 00:51:32,730 --> 00:51:35,180 That's what it's saying. 632 00:51:35,180 --> 00:51:37,455 And that was the content of quadratic variation, 633 00:51:37,455 --> 00:51:38,840 if you remember. 634 00:51:38,840 --> 00:51:42,484 We're summing the squares-- maybe not exactly this, 635 00:51:42,484 --> 00:51:44,650 but you're summing the squares over small intervals. 636 00:52:00,530 --> 00:52:02,510 So that's a really good fact that you can 637 00:52:02,510 --> 00:52:05,900 use to compute the variance. 638 00:52:05,900 --> 00:52:08,220 You have an Ito integral, you know the square, 639 00:52:08,220 --> 00:52:10,190 can be computed this simple way. 640 00:52:14,110 --> 00:52:15,090 That's really cool. 641 00:52:17,850 --> 00:52:19,080 And one more property. 642 00:52:19,080 --> 00:52:22,560 This one will be really important. 643 00:52:22,560 --> 00:52:24,295 You'll see it a lot in future lectures. 644 00:52:28,630 --> 00:52:31,190 It's that when is Ito integral a martingale? 645 00:52:46,630 --> 00:52:48,030 What's a martingale? 646 00:52:48,030 --> 00:52:52,310 Martingale meant if you have a stochastic process, 647 00:52:52,310 --> 00:53:01,080 at any time t, whatever happens after that, the expected value 648 00:53:01,080 --> 00:53:03,890 at time t is equal to 0. 649 00:53:03,890 --> 00:53:07,630 It doesn't have any natural tendency to go up or go down. 650 00:53:07,630 --> 00:53:10,357 No matter which point you stop your process 651 00:53:10,357 --> 00:53:12,815 and you see your future, it doesn't have a natural tendency 652 00:53:12,815 --> 00:53:15,470 to go up or go down. 653 00:53:15,470 --> 00:53:29,190 In formal language, it can be defined as where F_t 654 00:53:29,190 --> 00:53:32,670 is the events X_0 up to X_t. 655 00:53:35,890 --> 00:53:39,610 So if you take the conditional expectation 656 00:53:39,610 --> 00:53:42,300 based on whatever happened up to time t, 657 00:53:42,300 --> 00:53:44,235 that expectation will just be whatever value 658 00:53:44,235 --> 00:53:45,384 you have at that time. 659 00:53:48,524 --> 00:53:51,190 Intuitively, that just means you don't have any natural tendency 660 00:53:51,190 --> 00:53:53,900 to go up or go down. 661 00:53:53,900 --> 00:53:59,470 Question is, when is an Ito integral a martingale? 662 00:54:28,985 --> 00:54:35,710 Adapted to B of t, then... 663 00:54:45,344 --> 00:54:46,010 is a martingale. 664 00:54:51,030 --> 00:54:54,090 As long as g is not some crazy function, 665 00:54:54,090 --> 00:55:05,392 as long as g is reasonable-- one way can be reasonable if its 666 00:55:05,392 --> 00:55:07,900 L^2-norm is bounded. 667 00:55:07,900 --> 00:55:11,540 If you don't know what it means, you can safely ignore it. 668 00:55:19,030 --> 00:55:23,490 Basically, if g doesn't-- it's not a crazy function if it 669 00:55:23,490 --> 00:55:27,800 doesn't grow too fast, then in most cases this integral is 670 00:55:27,800 --> 00:55:28,960 always a martingale. 671 00:55:31,590 --> 00:55:34,800 If you flip it-- remember, integral 672 00:55:34,800 --> 00:55:38,880 was defined as the inverse of differentiation. 673 00:55:38,880 --> 00:55:42,700 So if dX_t is equal to some function mu, that 674 00:55:42,700 --> 00:55:48,631 depends on both t and B_t, times dt, plus sigma 675 00:55:48,631 --> 00:55:56,940 of dB_t, what this means is X_t is a martingale 676 00:55:56,940 --> 00:56:02,690 if that is 0 at all time, always. 677 00:56:07,860 --> 00:56:09,410 And if it's not 0, you have a drift, 678 00:56:09,410 --> 00:56:12,132 so it's not a martingale. 679 00:56:12,132 --> 00:56:13,590 That gives you some classification. 680 00:56:13,590 --> 00:56:15,390 Now, if you look at a differential equation 681 00:56:15,390 --> 00:56:17,310 of this stochastic-- this is called 682 00:56:17,310 --> 00:56:19,791 a stochastic differential equation-- if you know 683 00:56:19,791 --> 00:56:22,290 stochastic process, if you look at a stochastic differential 684 00:56:22,290 --> 00:56:26,440 equation, if it doesn't have a drift term, it's a martingale. 685 00:56:26,440 --> 00:56:29,510 If it has a drift term, it's not a martingale. 686 00:56:29,510 --> 00:56:32,310 That'll be really useful later, so try to remember it. 687 00:56:32,310 --> 00:56:34,290 The whole point is when you write down 688 00:56:34,290 --> 00:56:36,990 a stochastic process in terms of something times dt, 689 00:56:36,990 --> 00:56:40,200 something times dB_t, really this term 690 00:56:40,200 --> 00:56:45,640 contributes towards the tendency, the slope of whatever 691 00:56:45,640 --> 00:56:47,320 is going to happen in the future. 692 00:56:47,320 --> 00:56:50,595 And this is like the variance term. 693 00:56:50,595 --> 00:56:54,430 It adds some variance to your stochastic process. 694 00:56:54,430 --> 00:56:58,990 But still, it doesn't add or subtract value over time, 695 00:56:58,990 --> 00:57:03,895 it fairly adds variation. 696 00:57:06,540 --> 00:57:07,150 Remember that. 697 00:57:07,150 --> 00:57:09,890 That's very important fact. 698 00:57:09,890 --> 00:57:11,900 You're going to use it a lot. 699 00:57:11,900 --> 00:57:14,430 For example, you're going to use it for pricing theory. 700 00:57:14,430 --> 00:57:18,870 In pricing theory, you come up with this stochastic process 701 00:57:18,870 --> 00:57:20,150 or some strategy. 702 00:57:20,150 --> 00:57:22,130 You look at its value. 703 00:57:22,130 --> 00:57:27,280 Let's say X_t is your value of your portfolio over time. 704 00:57:27,280 --> 00:57:34,940 If that portfolio has-- then you match it with your financial-- 705 00:57:34,940 --> 00:57:36,830 let me go over it slowly again. 706 00:57:36,830 --> 00:57:44,780 First you have a financial derivative, like option 707 00:57:44,780 --> 00:57:47,632 of a stock. 708 00:57:47,632 --> 00:57:49,215 Then you have your portfolio strategy. 709 00:57:55,630 --> 00:57:57,720 Assume that you have some strategy that, 710 00:57:57,720 --> 00:57:59,940 at the expiration time, gives you 711 00:57:59,940 --> 00:58:01,310 the exact value of the option. 712 00:58:03,820 --> 00:58:05,820 Now you look at the difference between these two 713 00:58:05,820 --> 00:58:06,695 stochastic processes. 714 00:58:10,940 --> 00:58:14,880 Basically what the thing is, when your variance goes to 0, 715 00:58:14,880 --> 00:58:19,310 your drift also has to go to 0. 716 00:58:19,310 --> 00:58:20,920 So when you look at the difference, 717 00:58:20,920 --> 00:58:24,010 if you can somehow get rid of this variance term, that 718 00:58:24,010 --> 00:58:26,880 means no matter what you do, that 719 00:58:26,880 --> 00:58:30,660 will govern the value of your portfolio. 720 00:58:30,660 --> 00:58:34,084 If it's positive, that means you can always make money, 721 00:58:34,084 --> 00:58:35,250 because there's no variance. 722 00:58:35,250 --> 00:58:37,280 Without variance, you make money. 723 00:58:37,280 --> 00:58:41,770 That's called arbitrage, and you cannot have that. 724 00:58:41,770 --> 00:58:43,980 But I won't go into further detail 725 00:58:43,980 --> 00:58:46,870 because Vasily will cover it next time. 726 00:58:46,870 --> 00:58:49,070 But just remember that flavor. 727 00:58:49,070 --> 00:58:51,820 So when you write something down in a stochastic differential 728 00:58:51,820 --> 00:58:55,790 equation form, that term is a drift term, 729 00:58:55,790 --> 00:58:57,272 that term is a variance term. 730 00:58:57,272 --> 00:58:59,230 And if you don't have drift, it's a martingale. 731 00:59:01,876 --> 00:59:03,290 That is very important. 732 00:59:12,290 --> 00:59:12,950 Any questions? 733 00:59:12,950 --> 00:59:16,658 That's kind of the basics of Ito calculus. 734 00:59:22,520 --> 00:59:26,260 I will give you some exercises on it, 735 00:59:26,260 --> 00:59:29,520 mostly just basic computation exercises, so that you'll 736 00:59:29,520 --> 00:59:31,120 get familiar with it. 737 00:59:31,120 --> 00:59:33,210 Try to practice it. 738 00:59:33,210 --> 00:59:38,320 And let me cover one more thing called Girsanov theorem. 739 00:59:38,320 --> 00:59:40,900 It's related, but these are really 740 00:59:40,900 --> 00:59:42,750 basics of the Ito calculus, so if you 741 00:59:42,750 --> 00:59:46,300 have any questions on this, please ask me 742 00:59:46,300 --> 00:59:48,724 right now before I move on to the next topic. 743 00:59:56,842 --> 00:59:58,840 The last thing I want to talk about today. 744 01:00:43,710 --> 01:00:47,007 Here is an underlying question. 745 01:00:47,007 --> 01:00:48,590 Suppose you have two Brownian motions. 746 01:00:57,050 --> 01:00:58,240 This is without drift. 747 01:01:01,810 --> 01:01:06,905 And you have another B tilde, Brownian motion with drift. 748 01:01:12,910 --> 01:01:15,530 These are two probability distributions over paths. 749 01:01:18,290 --> 01:01:21,320 According to B_t, you're more likely to have 750 01:01:21,320 --> 01:01:25,770 some Brownian motion that has no drift. 751 01:01:25,770 --> 01:01:28,290 That's a sample path. 752 01:01:28,290 --> 01:01:31,090 According to B tilde, you have some drift. 753 01:01:34,890 --> 01:01:37,533 Your Brownian motion will-- 754 01:01:41,240 --> 01:01:46,190 A typical path will follow this line and will follow that line. 755 01:01:46,190 --> 01:01:52,920 The question is this-- can we switch 756 01:01:52,920 --> 01:01:55,490 from this distribution to this distribution 757 01:01:55,490 --> 01:01:56,561 by a change of measure? 758 01:02:02,250 --> 01:02:12,730 Can we switch between the two measures 759 01:02:12,730 --> 01:02:23,720 to probability distributions by a change of measure? 760 01:02:30,990 --> 01:02:34,114 Let me go a little bit more what it really means. 761 01:02:34,114 --> 01:02:36,280 Assume that you're just looking at a Brownian motion 762 01:02:36,280 --> 01:02:41,760 from time 0 up to time t, some fixed time interval. 763 01:02:41,760 --> 01:02:47,610 Then according to B_t, let's say this is a sample path omega. 764 01:02:47,610 --> 01:02:54,060 You have some probability of omega-- this is a p.d.f. 765 01:02:54,060 --> 01:03:01,830 given by this Brownian motion B. And then you 766 01:03:01,830 --> 01:03:06,100 have another p.d.f., P tilde of omega, which is a p.d.f. 767 01:03:06,100 --> 01:03:11,240 given by B of t. 768 01:03:11,240 --> 01:03:14,990 The question is, does there exist a Z 769 01:03:14,990 --> 01:03:19,610 depending on omega such that P of omega 770 01:03:19,610 --> 01:03:23,740 is equal to Z times P tilde? 771 01:03:39,298 --> 01:03:40,589 Do you understand the question? 772 01:03:46,220 --> 01:03:49,420 Clearly, if you just look at it, they're quite different. 773 01:03:49,420 --> 01:03:52,220 The path that you get according to the distributions 774 01:03:52,220 --> 01:03:55,230 are quite different. 775 01:03:55,230 --> 01:03:57,740 It's not clear why we should expect it at all. 776 01:04:02,866 --> 01:04:03,990 You'll see the answer soon. 777 01:04:03,990 --> 01:04:07,260 But let me discuss all this in a different context. 778 01:04:17,080 --> 01:04:19,880 Just forget about all the Brownian motion and everything 779 01:04:19,880 --> 01:04:22,410 just for a moment. 780 01:04:22,410 --> 01:04:26,010 In this concept, changing from one probability distribution 781 01:04:26,010 --> 01:04:29,270 to another distribution, it's a very important concept 782 01:04:29,270 --> 01:04:32,930 in analysis and probability just in general, theoretically. 783 01:04:32,930 --> 01:04:39,060 And there's a name for this Z, for this changing measure. 784 01:04:39,060 --> 01:04:46,380 If Z exists, it's called the Radon-Nikodym derivative. 785 01:04:50,900 --> 01:04:53,096 Before doing that, let me talk a little bit more. 786 01:04:59,940 --> 01:05:03,910 Suppose P is a probability distribution over omega. 787 01:05:09,204 --> 01:05:10,537 It's a probability distribution. 788 01:05:18,660 --> 01:05:21,560 So this is some set, and P describes the probability 789 01:05:21,560 --> 01:05:25,660 that you have each element in the set. 790 01:05:25,660 --> 01:05:27,970 And you have another probability distribution, P tilde. 791 01:05:33,520 --> 01:05:45,760 We define P and P tilde to be equivalent if the probability 792 01:05:45,760 --> 01:05:50,667 that A is greater than zero if and only if... 793 01:05:53,800 --> 01:05:54,400 For all... 794 01:05:58,310 --> 01:06:01,150 These probability distributions describe the probability 795 01:06:01,150 --> 01:06:03,380 of the subsets. 796 01:06:03,380 --> 01:06:05,210 Think about a very simple case. 797 01:06:05,210 --> 01:06:09,620 Sigma is equal to 1, 2, and 3. 798 01:06:09,620 --> 01:06:13,495 P gives 1/3 probability to 1, 1/3 probability 799 01:06:13,495 --> 01:06:16,970 to 2, 1/3 probability to 3. 800 01:06:16,970 --> 01:06:22,660 P tilde gives 2/3 probability to 3, 1 over 6 probability 801 01:06:22,660 --> 01:06:26,570 to 2, 1 over 6 probability to 3. 802 01:06:26,570 --> 01:06:29,290 We have two probability distribution over some space. 803 01:06:32,020 --> 01:06:34,790 They are equivalent if, whenever you 804 01:06:34,790 --> 01:06:39,210 take a subset of your background set-- let's say 1, 2. 805 01:06:39,210 --> 01:06:41,560 When A is equal to 1, 2, according 806 01:06:41,560 --> 01:06:44,810 to probability distribution P, the probability 807 01:06:44,810 --> 01:06:48,125 you fall into this set A is equal to 2/3. 808 01:06:50,770 --> 01:06:54,750 According to P tilde, you have 5/6. 809 01:06:57,460 --> 01:06:58,590 They're not the same. 810 01:06:58,590 --> 01:07:00,460 The probability itself is not the same, 811 01:07:00,460 --> 01:07:03,220 but this condition is satisfied when it's 0. 812 01:07:03,220 --> 01:07:04,865 And when it's not 0, it's not 0. 813 01:07:04,865 --> 01:07:07,406 And you can just check that it's always true, because they're 814 01:07:07,406 --> 01:07:09,360 all positive probabilities. 815 01:07:09,360 --> 01:07:14,270 On the other hand, if you take instead, say, 816 01:07:14,270 --> 01:07:19,640 1/3 and 0, now you take your A to be 3. 817 01:07:22,850 --> 01:07:28,610 Then you have 1/3 equal to 0. 818 01:07:28,610 --> 01:07:33,160 This means, according to probability distribution P, 819 01:07:33,160 --> 01:07:37,320 there is some probability that you'll get 3. 820 01:07:37,320 --> 01:07:39,990 But according to probability distribution P tilde, 821 01:07:39,990 --> 01:07:43,970 you don't have any probability of getting 3. 822 01:07:43,970 --> 01:07:47,620 So they're not equivalent in this case. 823 01:07:47,620 --> 01:07:49,840 If you think about it, then it's really clear. 824 01:07:49,840 --> 01:07:52,360 The theorem says-- this is a very important theorem 825 01:07:52,360 --> 01:07:53,690 in analysis, actually. 826 01:07:57,875 --> 01:08:08,475 The theorem-- there exists a Z such that P of omega is equal 827 01:08:08,475 --> 01:08:08,975 to... 828 01:08:12,380 --> 01:08:16,638 If and only if P and P tilde are equivalent. 829 01:08:22,510 --> 01:08:24,750 You can change from one probability measure 830 01:08:24,750 --> 01:08:27,029 to another probability measure just 831 01:08:27,029 --> 01:08:32,231 in terms of multiplication, if and only if they're equivalent. 832 01:08:32,231 --> 01:08:34,189 And you can see that it's not the case for this 833 01:08:34,189 --> 01:08:35,355 when they're not equivalent. 834 01:08:35,355 --> 01:08:37,740 You can't make a zero probability to 1/3 probability 835 01:08:37,740 --> 01:08:40,000 by multiplication. 836 01:08:40,000 --> 01:08:44,510 So in the finite world this is very just intuitive theorem, 837 01:08:44,510 --> 01:08:48,490 but what this is saying is it's true for all probability 838 01:08:48,490 --> 01:08:50,736 spaces. 839 01:08:50,736 --> 01:08:52,819 And these are called the Radon-Nikodym derivative. 840 01:09:01,930 --> 01:09:06,990 Our question is, are these two Brownian motions equivalent? 841 01:09:06,990 --> 01:09:09,915 The paths that this Brownian motion without drift 842 01:09:09,915 --> 01:09:12,330 takes and the Brownian motion with drift 843 01:09:12,330 --> 01:09:16,529 takes, are they kind of the same but just skewed 844 01:09:16,529 --> 01:09:20,562 in distribution, or are they really fundamentally different? 845 01:09:20,562 --> 01:09:21,538 That's the question. 846 01:09:28,859 --> 01:09:33,479 And what Girsanov's theorem says is that they are equivalent. 847 01:09:33,479 --> 01:09:36,089 To me, it came as a little bit non-intuitive. 848 01:09:36,089 --> 01:09:39,880 I would imagine that it's not equivalent, these two. 849 01:09:39,880 --> 01:09:42,069 These paths have a very natural tendency. 850 01:09:42,069 --> 01:09:44,870 As it goes to infinity, these paths and these paths 851 01:09:44,870 --> 01:09:47,779 will really look a lot different, 852 01:09:47,779 --> 01:09:51,000 because when you go really, really far, 853 01:09:51,000 --> 01:09:53,939 the paths which have drift will be just really 854 01:09:53,939 --> 01:09:57,870 close to your line mu of t, while the paths which 855 01:09:57,870 --> 01:10:00,302 don't have drift will be really close to the x-axis. 856 01:10:02,900 --> 01:10:06,070 But still, they are equivalent. 857 01:10:06,070 --> 01:10:09,590 You can change from one to another. 858 01:10:09,590 --> 01:10:13,840 I'll just state that theorem without proof. 859 01:10:13,840 --> 01:10:17,345 And this will also be used in pricing theory. 860 01:10:20,930 --> 01:10:23,270 I'm not an expert enough to tell why, 861 01:10:23,270 --> 01:10:25,140 but basically what it's saying is, 862 01:10:25,140 --> 01:10:28,580 you switch some stochastic process 863 01:10:28,580 --> 01:10:31,000 into a stochastic process without drift, 864 01:10:31,000 --> 01:10:33,610 thus making it into a martingale. 865 01:10:33,610 --> 01:10:36,180 And martingale has a lot of meaning in pricing theory, 866 01:10:36,180 --> 01:10:38,310 as you'll see. 867 01:10:38,310 --> 01:10:39,920 This also has application. 868 01:10:39,920 --> 01:10:42,030 That's why I'm trying to cover it, although it's 869 01:10:42,030 --> 01:10:44,300 quite a technical theorem. 870 01:10:44,300 --> 01:10:46,985 Try to remember at least a statement and the spirit 871 01:10:46,985 --> 01:10:48,700 of what it means. 872 01:10:48,700 --> 01:10:50,690 It just means these two are equivalent, 873 01:10:50,690 --> 01:10:52,370 you can change from one to another 874 01:10:52,370 --> 01:10:53,830 by a multiplicative function. 875 01:11:08,267 --> 01:11:09,850 Let me just state it in a simple form. 876 01:11:12,615 --> 01:11:14,740 GUEST SPEAKER: If I could just interject a comment. 877 01:11:14,740 --> 01:11:15,406 PROFESSOR: Sure. 878 01:11:15,406 --> 01:11:19,830 GUEST SPEAKER: With these changes of measure, 879 01:11:19,830 --> 01:11:24,620 it turns out that all of these theories with continuous time 880 01:11:24,620 --> 01:11:27,530 processes should have an interpretation if you've 881 01:11:27,530 --> 01:11:30,720 discretized time, and should consider 882 01:11:30,720 --> 01:11:34,140 sort of a finer and finer discretization of the process. 883 01:11:34,140 --> 01:11:40,680 And with this change of measure, if you consider problems 884 01:11:40,680 --> 01:11:45,840 in discrete stochastic processes like random walks, 885 01:11:45,840 --> 01:11:52,160 basically how-- say if you're gambling against a casino 886 01:11:52,160 --> 01:11:54,510 or against another player, and you 887 01:11:54,510 --> 01:11:58,200 look at how your winnings evolve as a random walk, 888 01:11:58,200 --> 01:11:59,780 depending on your odds, your odds 889 01:11:59,780 --> 01:12:03,340 could be that you will tend to lose. 890 01:12:03,340 --> 01:12:06,740 So there's basically a drift in your wealth 891 01:12:06,740 --> 01:12:08,940 as this random process evolves. 892 01:12:08,940 --> 01:12:15,820 You can transform that process, basically by taking out 893 01:12:15,820 --> 01:12:19,560 your expected losses, to a process which 894 01:12:19,560 --> 01:12:22,830 has zero change in expectation. 895 01:12:22,830 --> 01:12:26,840 And so you can convert these gambling problems 896 01:12:26,840 --> 01:12:32,020 where there's drift to a version where the process, essentially, 897 01:12:32,020 --> 01:12:34,460 has no drift and is a martingale. 898 01:12:34,460 --> 01:12:37,240 And the martingale theory in stochastic process courses 899 01:12:37,240 --> 01:12:38,560 is very, very powerful. 900 01:12:38,560 --> 01:12:41,090 There's martingale convergence theorems. 901 01:12:41,090 --> 01:12:44,180 So you know that the limit of the martingale 902 01:12:44,180 --> 01:12:48,750 is-- there's a convergence of the process, 903 01:12:48,750 --> 01:12:50,530 and that applies here as well. 904 01:12:55,026 --> 01:12:57,234 PROFESSOR: You will see some surprising applications. 905 01:12:57,234 --> 01:12:59,594 GUEST SPEAKER: Yeah. 906 01:12:59,594 --> 01:13:03,515 PROFESSOR: And try to at least digest the statement. 907 01:13:08,540 --> 01:13:12,340 When the guest speaker comes and says by Girsanov theorem, 908 01:13:12,340 --> 01:13:15,660 they actually know what it is. 909 01:13:15,660 --> 01:13:16,410 There's a spirit. 910 01:13:20,190 --> 01:13:21,707 This is a very simple version. 911 01:13:21,707 --> 01:13:23,290 There's a lot of complicated versions, 912 01:13:23,290 --> 01:13:24,874 but let me just do it. 913 01:13:30,570 --> 01:13:40,025 So P is a probability distribution over paths 914 01:13:40,025 --> 01:13:41,900 from [0, T] to the infinity. 915 01:13:41,900 --> 01:13:48,860 What this means is just paths from that-- stochastic process 916 01:13:48,860 --> 01:13:53,110 defined from time 0 to time T. These 917 01:13:53,110 --> 01:14:10,790 are paths defined by a Brownian motion with drift mu. 918 01:14:10,790 --> 01:14:14,080 And then P tilde is a probability distribution 919 01:14:14,080 --> 01:14:16,821 defined by Brownian motion without drift. 920 01:14:22,600 --> 01:14:27,445 Then P and P tilde are equivalent. 921 01:14:27,445 --> 01:14:29,320 Not only are they equivalent, we can actually 922 01:14:29,320 --> 01:14:31,530 compute their Radon-Nikodym derivative. 923 01:14:36,184 --> 01:14:44,378 And the Radon-Nikodym derivative Z 924 01:14:44,378 --> 01:14:50,100 which is defined as T of-- which we denote like this 925 01:14:50,100 --> 01:14:51,180 has this nice form. 926 01:15:05,920 --> 01:15:08,150 That's a nice closed form. 927 01:15:08,150 --> 01:15:13,120 Let me just tell you a few implications of this. 928 01:15:31,490 --> 01:15:35,790 Now, assume you have some, let's say, value 929 01:15:35,790 --> 01:15:37,430 of your portfolio over time. 930 01:15:37,430 --> 01:15:40,230 That's the stochastic process. 931 01:15:40,230 --> 01:15:44,090 And you measure it according to this probability distribution. 932 01:15:44,090 --> 01:15:45,930 Let's say it depends on some stock price 933 01:15:45,930 --> 01:15:47,970 as the stock price is modeled using a Brownian 934 01:15:47,970 --> 01:15:51,140 motion with drift. 935 01:15:51,140 --> 01:15:53,500 What this is saying is, now, instead 936 01:15:53,500 --> 01:15:57,920 of computing this expectation in your probability space-- 937 01:15:57,920 --> 01:16:03,140 so this is defined over the probability space P, 938 01:16:03,140 --> 01:16:06,510 our sigma-- (omega, P) defined by this probability 939 01:16:06,510 --> 01:16:07,610 distribution. 940 01:16:07,610 --> 01:16:25,730 You can instead compute it in-- you 941 01:16:25,730 --> 01:16:28,720 can compute as expectation in a different probability space. 942 01:16:35,080 --> 01:16:38,430 You transform the problems about Brownian motion with drift 943 01:16:38,430 --> 01:16:41,420 into a problem about Brownian motion without a drift. 944 01:16:41,420 --> 01:16:43,170 And the reason I have Z tilde instead of Z 945 01:16:43,170 --> 01:16:45,230 here is because I flipped. 946 01:16:45,230 --> 01:16:54,480 What you really should have is Z tilde here as expectation of Z. 947 01:16:54,480 --> 01:16:59,380 If you want to use this Z. 948 01:16:59,380 --> 01:17:03,350 I don't expect you to really be able to do computations 949 01:17:03,350 --> 01:17:06,650 and do that just by looking at this theorem once. 950 01:17:06,650 --> 01:17:09,384 Just really trying to digest what it means 951 01:17:09,384 --> 01:17:13,050 and understand the flavor of it, that you can transform 952 01:17:13,050 --> 01:17:14,650 problems in one probability space 953 01:17:14,650 --> 01:17:16,990 to another probability space. 954 01:17:16,990 --> 01:17:19,460 And you can actually do that when the two distributions are 955 01:17:19,460 --> 01:17:22,700 defined by Brownian motions when one has drift and one 956 01:17:22,700 --> 01:17:25,000 doesn't have a drift. 957 01:17:25,000 --> 01:17:27,700 How we're going to use it is we're 958 01:17:27,700 --> 01:17:29,802 going to transform a non-martingale process 959 01:17:29,802 --> 01:17:30,885 into a martingale process. 960 01:17:35,314 --> 01:17:36,730 When you change into martingale it 961 01:17:36,730 --> 01:17:39,662 has very good physical meanings to it. 962 01:17:43,450 --> 01:17:44,680 That's it for today. 963 01:17:44,680 --> 01:17:48,030 And you only have one more math lecture remaining 964 01:17:48,030 --> 01:17:51,580 and maybe one or two homeworks but if you have two, 965 01:17:51,580 --> 01:17:54,950 the second one won't be that long. 966 01:17:54,950 --> 01:17:57,340 And you'll have a lot of guest lectures, exciting guest 967 01:17:57,340 --> 01:18:00,990 lectures, so try not to miss them.